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\topic{Lecture 5 \\Differential Calculus-I\\ \scriptsize Partial Differentiation:Euler's Theorem (23 Sep 2009)}
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\subsection*{Homogeneous Functions}
The expression
\[a_0x^n + a_1x^{n-1}y+ a_2x^{n-2}y^2 + \dots + + a_{n-1}xy^{n-1} + a_n y^n \]
has every term is of degree $n$. Such expressions in which every degree if of same order, are called Hompgeneous functions of degree $n$. If we take $x^n$ as common, we may write it also as
\[x^n \left[a_0 + a_1\frac{y}{x} + a_2 \left(\frac{y}{x}\right)^2 + \dots + a_n \left(\frac{y}{x}\right)^n \right]\]
Hence the general definition of homogeneous function is as follows:
\begin{quote}
$x^nf(y/x)$ is called a homogeneous function of degree $n$, whatever be the function $f$. For example $x^3 \sin(y/x)$ is homogeneous function of degree $3$.
\end{quote}
\subsection*{Euler's Theorem}
\begin{quote}
If $f(x,y)$ be a homogeneous function of $x$ and $y$ of degree $n$, then\footnote{This theorem is also true for any number of variables} \[x \pd{f}{x} + y \pd{f}{y} = nf \]
\end{quote}
As $f(x,y)$ is homogeneous function of degree $n$, we may write
\[x^nf(\frac{y}{x})\]
Now,
\[
\begin{array}{rcl}
\pd{f}{x}  =&  nx^{n-1}f(\frac{y}{x}) & +  x^{n}f'(\frac{y}{x}).\frac{y}{-x^2} \\
\pd{f}{y}  =&  & +  x^{n}f'(\frac{y}{x}).\frac{1}{x}  \\
\end{array}
\]
Multiply first equation by $x$ and second by $y$
\[
\begin{array}{rcl}
x\pd{f}{x}  =&  nx^{n}f(\frac{y}{x}) & -  x^{n}f'(\frac{y}{x}).\frac{y}{x} \\
y\pd{f}{y}  =&  & +  x^{n}f'(\frac{y}{x}).\frac{y}{x}  \\
\end{array}
\]
On Adding, we get
\[
x\pd{f}{x}+y\pd{f}{y} = nx^{n}f(\frac{y}{x}) = nf(x,y) \
\]
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\begin{example}
Verify Euler's Theroem when $f(x,y,z) = 3x^2yz+5xy^2z+4z^3$
\end{example}
Here degree of $f$ is $4$, by Euler's Theorem, we have
\[xf_x+yf_y+zf_z = 4 f(x,y,z)\]
Here, $f_x = 6xyz+5y^2z$, $f_y = 3x^2z+10xyz$, $f_z = 3x^2y+5xy^2+12z^2$, 
Now, 
\[xf_x+yf_y+zf_z = (6x^2yz+5xy^2z) + (3x^2yz+10xy^2z) + (3x^2yz+5xy^2z+12z^3) \]
\[xf_x+yf_y+zf_z = 4(3x^2yz+5xy^2z+4z^3) = 4 f(x,y,z)\]
Which is true.Hence

\section*{Problems}
\begin{enumerate}
\item  Verify Euler's Theorem for the following
(a)	$f(x,y)=ax^2+2hxy+by^2$ (b) $u=\frac{x^{1/4}+y^{1/4}}{x^{1/5}+y^{1/5}}$
\item  If $u$ is a homogeneous function of degree $n$, show that 
			\[x \frac{\partial^2u}{\partial x^2 } + y \frac{\partial^2u}{\partial x \partial y} = (n-1) \pd{u}{x}\]
			\[x \frac{\partial^2u}{\partial x \partial y} + y \frac{\partial^2u}{\partial y^2 }  = (n-1) \pd{u}{y}\]
\item  If $u=\sin^{-1}(x/y)+tan^{-1}(y/x)$, prove that
\[x \pd{u}{x} + y \pd{u}{y} = 0 \]
\item  Deduce from Euler's theorem
			\[x^2 \frac{\partial^2u}{\partial x^2 } + 2xy \frac{\partial^2u}{\partial x \partial y} + y^2 \frac{\partial^2u}{\partial y^2 } = n(n-1)u\]
\item  If $u=\sin^{-1}\frac{x^2+y^2}{x+y}$, prove that
\[x \pd{u}{x} + y \pd{u}{y} = \tan u \]
\item  If $u=\tan^{-1}\frac{x^3+y^3}{x-y}$, prove that
\[x \pd{u}{x} + y \pd{u}{y} = \sin 2u \]
\end{enumerate}

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